Absolute Continuity

Last updated Nov 1, 2022

Let $(X, \mathcal{M})$ be a Measure Space and let $\mu, \nu$ be two Measures on this space. We say that $\nu « \mu$ ($\nu$ is absolute continuous wrt $\mu$) if $\forall A \in \mathcal{M}$ s.t. $\mu(A) = 0$, we have that $\nu(A) = 0$. TODO - update this for Signed Measures and Complex Measures

Some equivalent conditions:

1. A sigma-finite measure is absolutely continuous wrt another sigma-finite measure iff their Radon-Nikodym Derivative exists
2. A finite measure is absolutely continuous wrt another iff small sets on the other measure are also small on it